Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

Compute the sum of first n terms of the following series 6 + 66 + 666 + .......

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 4²) + (1 + 4 + 4² + 4³) + ...

Find the general terms and sum to n terms of the sequence 1, \(\frac{4}{3},\frac{7}{9},\frac{10}{27},....\)

Find the value of n if the sum to n terms of the series \(\sqrt { 3 } +\sqrt { 75 } +\sqrt { 243 } +....is\quad 435\sqrt { 3 } .\)

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

A man repays an amount of Rs. 3250 by paying Rs. 20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

In a race, 20 balls are placed in a line at intervals of 4 meters, with the first ball 24 meters away from the starting point. A contestant is required to bring the balls back to the starting place one at a time. How far would the contestant run to bring back all balls?

In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particle gets doubled each day, being 5 particles on the first day. Find the day when the infectious virus particles just grow over 1,50,000 units?

If x = 0.001, prove that \(\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } } \) = 8.01 up to two places of decimals 

If (p+1) ^th  term of an A.P is twice the (q+1)^th terms prove that the (3p+1)^th term is twice the  (p+q+1)^th term

Find the sum to n terms of the series 1 - 5 + 9 - 13+ ......

Expand \((x^2+\sqrt{1-x^2})^5+(x^2-\sqrt{1-x^2})^5.\)

Find \(\sqrt{x^2+4}-\sqrt{x^2-4}\) when x is large.

Evaluate \(\sum_{k=1}^{10}(k^2-3k+5)\).

Expand \({1\over (3+2x)^2}\) in powers of x. Find a condition on x for which the expansion is valid.

Prove that \(\sqrt [ 3 ]{ x^3+7 } -\sqrt [ 3 ]{ x^3+4 } \) is approximately equal to \({1\over x^2}\) when x is large.

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n, terms of a G.P. , then prove that S₁ (S₃ - S₂) = (S₂ - S₁)².

If sum of the n terms of a G.P be S, their product P and the sum of their reciprocals R, then prove that \(P^{2}=(\frac{S}{R})^{n}\)